A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory

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This is a textbook for an introductory combinatorics course lasting one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course.

Mikls Bna received his Ph.D. at Massachusetts institute of Technology. He is a Professor of Mathematics at the University of Florida, where he has been inducted into the Academy of Distinguished Teaching Scholars. His research has been supported by the National Science Foundation, the National Security Agency, and the Howard Hughes Medical institute. Mikls Bna has received teaching awards at the University of Florida and at the University of Pennsylvania. He is one of the Editor-in-Chiefs of the Electronic Journal of Combinatorics.

Foreword

p. vii

Preface

p. ix

Acknowledgments

p. xi

Basic Methods

Seven Is More Than Six. The Pigeon-Hole Principle

p. 1

The Basic Pigeon-Hole Principle

p. 1

The Generalized Pigeon-Hole Principle

p. 3

Exercises

p. 9

Supplementary Exercises

p. 11

Solutions to Exercises

p. 13

One Step at a Time. The Method of Mathematical Induction

p. 21

Weak Induction

p. 21

Strong Induction

p. 26

Exercises

p. 28

Supplementary Exercises

p. 30

Solutions to Exercises

p. 31

Enumerative Combinatorics

There Are A Lot Of Them. Elementary Counting Problems

p. 39

Permutations

p. 39

Strings over a Finite Alphabet

p. 42

Choice Problems

p. 45

Exercises

p. 49

Supplementary Exercises

p. 53

Solutions to Exercises

p. 55

No Matter How You Slice It. The Binomial Theorem and Related Identities